Page 188 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes p = (x – c ) ... (x – c )
1 k
where c ,..., c are distinct elements of F.
1 k
Proof: We have noted earlier that, if T is diagonalizable, its minimal polynomial is a product of
distinct linear factors. To prove the converse, let W be the subspace spanned by all of the
characteristic vectors of T, and suppose W  V. By the lemma used in the proof of Theorem 1,
there is a vector  not in W and a characteristic value c of T such that the vector
j
 = (T – c I)
j
lies in W. Since  is in W,
 =  + ... + 
1 k
where T = c  , 1  i  k, and therefore the vector
i i i
h(T) = h(c ) + ... + h(c )
1 1 k k
is in W, for every polynomial h.
Now p = (x – c )q, for some polynomial q. Also
j
q – q(c ) = (x – c )h
j j
We have
q(T) – q(c ) = h(T)(T – c I) = h(T’)
j j
But h(T) is in W and, since

0 = p(T) = (T – c I)q(T)
j
the vector q(T) is in W. Therefore, q(c ) is in W. Since  is not in W, we have q(c ) = 0. That
j j
contradicts the fact that p has distinct roots.
In addition to being an elegant result, Theorem 2 is useful in a computational way. Suppose we
have a linear operator T, represented by the matrix A in some ordered basis, and we wish to
know if T is diagonalizable. We compute the characteristic polynomial f. If we can factor f:
1 d
f  (x c )  (x c ) k d


1 k
we have two different methods for determining whether or not T is diagonalizable. One method
is to see whether (for each i) we can find d independent characteristic vectors associated with the
i
characteristic value c . The other method is to check whether or not (T – c I)  (T – c I) is the zero
i 1 k
operator.
Theorem 1 provides a different proof of the Cayley-Hamilton theorem. That theorem is easy for
a triangular matrix. Hence, via Theorem 1, we obtain the result for any matrix over an
algebraically closed field. Any field is a subfield of an algebraically closed field. If one knows
that result, one obtains a proof of the Cayley-Hamilton theorem for matrices over any field. If
we at least admit into our discussion the Fundamental Theorem of Algebra (the complex number
field is algebraically closed), then Theorem 1 provides a proof of the Cayley-Hamilton theorem
for complex matrices, and that proof is independent of the one which we gave earlier.

Self Assessment

2
1. Let T be the linear operator on R , the matrix of which in the standard ordered basis is
1
 1  
A   
 2 2 

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